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  <div class="section" id="numpy-random-negative-binomial">
<h1>numpy.random.negative_binomial<a class="headerlink" href="#numpy-random-negative-binomial" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.random.negative_binomial">
<code class="sig-prename descclassname">numpy.random.</code><code class="sig-name descname">negative_binomial</code><span class="sig-paren">(</span><em class="sig-param">n</em>, <em class="sig-param">p</em>, <em class="sig-param">size=None</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.negative_binomial" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw samples from a negative binomial distribution.</p>
<p>Samples are drawn from a negative binomial distribution with specified
parameters, <em class="xref py py-obj">n</em> successes and <em class="xref py py-obj">p</em> probability of success where <em class="xref py py-obj">n</em>
is &gt; 0 and <em class="xref py py-obj">p</em> is in the interval [0, 1].</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <code class="docutils literal notranslate"><span class="pre">negative_binomial</span></code> method of a <code class="docutils literal notranslate"><span class="pre">default_rng()</span></code>
instance instead; see <em class="xref py py-obj">random-quick-start</em>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>n</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Parameter of the distribution, &gt; 0.</p>
</dd>
<dt><strong>p</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Parameter of the distribution, &gt;= 0 and &lt;=1.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">p</span></code> are both scalars.
Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(n,</span> <span class="pre">p).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized negative binomial distribution,
where each sample is equal to N, the number of failures that
occurred before a total of n successes was reached.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.negative_binomial.html#numpy.random.Generator.negative_binomial" title="numpy.random.Generator.negative_binomial"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator.negative_binomial</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability mass function of the negative binomial distribution is</p>
<div class="math">
<p><img src="../../../_images/math/0f0e0840eadc4e9dceb2e519e01553b7adf9272d.svg" alt="P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},"/></p>
</div><p>where <img class="math" src="../../../_images/math/5a939c5280da7202ca4531f175a7780ad5e1f80a.svg" alt="n"/> is the number of successes, <img class="math" src="../../../_images/math/141bbefb74014fc5e43499901bf78607ae335583.svg" alt="p"/> is the
probability of success, <img class="math" src="../../../_images/math/76328e665603d93f506fa36da6742d13b886f4b6.svg" alt="N+n"/> is the number of trials, and
<img class="math" src="../../../_images/math/e7003fd3463f843ee1e53385878369f078d362ad.svg" alt="\Gamma"/> is the gamma function. When <img class="math" src="../../../_images/math/5a939c5280da7202ca4531f175a7780ad5e1f80a.svg" alt="n"/> is an integer,
<img class="math" src="../../../_images/math/2e3f2812bc80c49cc2cbe20dbe00c9075572a2e4.svg" alt="\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}"/>, which is
the more common form of this term in the the pmf. The negative
binomial distribution gives the probability of N failures given n
successes, with a success on the last trial.</p>
<p>If one throws a die repeatedly until the third time a “1” appears,
then the probability distribution of the number of non-“1”s that
appear before the third “1” is a negative binomial distribution.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r09f005d8254d-1"><span class="brackets">1</span></dt>
<dd><p>Weisstein, Eric W. “Negative Binomial Distribution.” From
MathWorld–A Wolfram Web Resource.
<a class="reference external" href="http://mathworld.wolfram.com/NegativeBinomialDistribution.html">http://mathworld.wolfram.com/NegativeBinomialDistribution.html</a></p>
</dd>
<dt class="label" id="r09f005d8254d-2"><span class="brackets">2</span></dt>
<dd><p>Wikipedia, “Negative binomial distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Negative_binomial_distribution">https://en.wikipedia.org/wiki/Negative_binomial_distribution</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Draw samples from the distribution:</p>
<p>A real world example. A company drills wild-cat oil
exploration wells, each with an estimated probability of
success of 0.1.  What is the probability of having one success
for each successive well, that is what is the probability of a
single success after drilling 5 wells, after 6 wells, etc.?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">negative_binomial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">):</span> 
<span class="gp">... </span>   <span class="n">probability</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="o">&lt;</span><span class="n">i</span><span class="p">)</span> <span class="o">/</span> <span class="mf">100000.</span>
<span class="gp">... </span>   <span class="nb">print</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s2">&quot;wells drilled, probability of one success =&quot;</span><span class="p">,</span> <span class="n">probability</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>

</div>


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